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Marvel temperaments: Difference between revisions

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This page discusses some of the temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the [[Marvel family|marvel]] comma or septimal kleisma. These include negri, wizard, tritonic, septimin, slender, triton, merman and marvo. Considered elsewhere are [[Meantone family #Septimal meantone|meantone]], [[Gamelismic clan #Miracle|miracle]], [[Magic family|magic]], [[Diaschismic family #Pajara|pajara]], [[orwell]], [[Kleismic family #Catakleismic|catakleismic]], [[Schismatic family #Garibaldi|garibaldi]], [[Augmented family #August|august]], [[Pythagorean family #Compton|compton]], [[Dicot family #Sharp|sharp]], [[Escapade family #Escapade|escapade]], [[Qintosec family|qintosec]], [[Pelogic family #Mavila|mavila]] and [[Cloudy clan #Decic|decic]].
{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.  


Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
Temperaments considered in families and clans are:
* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
 
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]].
 
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.


The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.


Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)


= Negri =
== Wizard ==
{{main| Negri }}
{{Main| Wizard }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''


Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.  
Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.  


== 5-limit ==
[[Subgroup]]: 2.3.5.7


Subgroup: 2.3.5
[[Comma list]]: 225/224, 118098/117649


[[Comma list]]: 16875/16384
{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
: mapping generators: ~1225/864, ~245/216


[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
[[Optimal tuning]]s:  
* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}


{{Multival|legend=1| 4 -3 -14 }}
{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}


[[POTE generator]]: ~16/15 = 125.7549
[[Badness]] (Sintel): 1.03


{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


== 7-limit ==
Comma list: 225/224, 385/384, 4000/3993


Subgroup: 2.3.5.7
Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}


[[Comma list]]: 49/48, 225/224
Optimal tunings:  
* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}


{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
Badness (Sintel): 0.613


[[POTE generator]]: ~15/14 = 125.608
==== Lizard ====
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
Comma list: 225/224, 351/350, 364/363, 385/384


=== Negra ===
Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
This is the 2.3.5.7.13 extension of negri.


Subgroup: 2.3.5.7.13
Optimal tunings:  
* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}


[[Comma list]]: 49/48, 65/64, 91/90
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


[[Sval]] [[Mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
Badness (Sintel): 0.900


Gencom mapping: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374


POTE generator: ~14/13 = 125.567
Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}


{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
Optimal tunings:
* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 22, 50, 72 }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.741


Comma list: 45/44, 49/48, 56/55
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342


POTE generator: ~15/14 = 126.474
Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}


{{Val list|legend=1| 9, 10, 19 }}
Optimal tunings:
* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}


Badness: 0.0262
{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}


</div></div>
Badness (Sintel): 0.955
 
=== 13-limit ===


==== Gizzard ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 49/48, 56/55, 78/77
Comma list: 225/224, 325/324, 385/384, 1573/1568


Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}


POTE generator: ~14/13 = 126.431
Optimal tunings:  
* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}


{{Val list|legend=1| 9, 10, 19 }}
{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}


== Negril ==
Badness (Sintel): 0.837


Subgroup: 2.3.5.7.11
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 100/99, 225/224
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384


Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}


POTE generator: ~15/14 = 124.767
Optimal tunings:  
* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}


{{Val list|legend=1| 19, 29, 48d, 77cdd }}
{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}


Badness: 0.0387
Badness (Sintel): 0.694


=== 13-limit ===
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594


Comma list: 49/48, 65/64, 91/90, 875/858
Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}


Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
Optimal tunings:  
* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}


POTE generator: ~14/13 = 124.716
{{Optimal ET sequence|legend=0| 72, 94, 166g }}


{{Val list|legend=1| 19, 29, 48df, 77cddf }}
Badness (Sintel): 0.901
 
Badness: 0.0244
 
== Negric ==


=== Mage ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 33/32, 49/48, 77/75
Comma list: 99/98, 176/175, 1331/1296


Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}


POTE generator: ~15/14 = 127.039
Optimal tunings:  
* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}


Badness: 0.0306
Badness (Sintel): 1.91


=== 13-limit ===
== Tritonic ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''


Subgroup: 2.3.5.7.11.13
Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.  


Comma list: 33/32, 49/48, 65/64, 91/90
[[Subgroup]]: 2.3.5.7


Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
[[Comma list]]: 225/224, 50421/50000


POTE generator: ~14/13 = 127.039
{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
: mapping generators: ~2, ~10/7


{{Val list|legend=1| 9, 19e, 47…, 66…, 85… }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}


Badness: 0.0202
{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}


== Negroni ==
[[Badness]] (Sintel): 1.20


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 55/54, 225/224
Comma list: 121/120, 225/224, 441/440


Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}


POTE generator: ~15/14 = 124.539
Optimal tunings:  
* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}


{{Val list|legend=1| 10, 19e, 29, 77cddee }}
{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}


Badness: 0.0353
Badness (Sintel): 0.782
 
=== 13-limit ===


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 55/54, 65/64, 91/90
Comma list: 105/104, 121/120, 196/195, 275/273


Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}


POTE generator: ~14/13 = 124.545
Optimal tunings:  
* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}


{{Val list|legend=1| 10, 19e, 29, 77cddeef }}
{{Optimal ET sequence|legend=0| 29, 31, 60e }}


Badness: 0.0216
Badness (Sintel): 0.950


== Wilsec ==
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7.11
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272


Comma list: 49/48, 121/120, 225/224
Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}


Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
Optimal tunings:  
* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}


POTE generator: ~11/8 = 537.186
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


{{Val list|legend=1| 9, 20, 29, 38d, 67cdde }}
Badness (Sintel): 0.973


Badness: 0.0419
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


=== 13-limit ===
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272


Subgroup: 2.3.5.7.11.13
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}


Comma list: 49/48, 65/64, 91/90, 121/120
Optimal tunings:  
* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}


Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


POTE generator: ~11/8 = 537.208
Badness (Sintel): 1.03


{{Val list|legend=1| 9, 20, 29, 38df, 67cddef }}
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


Badness: 0.0252
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195


=== 17-limit ===
Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}


Subgroup: 2.3.5.7.11.13.17
Optimal tunings:  
* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}


Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
{{Optimal ET sequence|legend=0| 29g, 31, 60e }}


Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
Badness (Sintel): 1.04


POTE generator: ~11/8 = 537.230
=== Tritoni ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefg }}
Comma list: 225/224, 385/384, 27783/27500


Badness: 0.0218
Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}


=== 19-limit ===
Optimal tunings:
* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}


Subgroup: 2.3.5.7.11.13.17.19
{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}


Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
Badness (Sintel): 1.50


Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
== Septimin ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''


POTE generator: ~11/8 = 537.214
Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.  


{{Val list|legend=1| 9, 20g, 29g, 38df, 67cddefgh }}
[[Subgroup]]: 2.3.5.7


Badness: 0.0168
[[Comma list]]: 225/224, 84035/82944


= Passive =
{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
{{see also | Archytas clan #Passion }}
: mapping generators: ~2, ~12/7


Subgroup: 2.3.5.7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}


[[Comma list]]: 225/224, 256/245
{{Optimal ET sequence|legend=1| 41, 91, 132d }}


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -5 4 -2 }}]
[[Badness]] (Sintel): 1.38


[[POTE generator]]: ~16/15 = 98.809
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 12, 49dd }}
Comma list: 225/224, 245/242, 385/384


[[Badness]]: 0.0751
Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}


= Wizard =
Optimal tunings:
* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
{{See also|Wizard}}


Subgroup: 2.3.5.7
Badness (Sintel): 1.04


[[Comma list]]: 225/224, 118098/117649
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
Comma list: 105/104, 144/143, 196/195, 245/242


Mapping generators: ~1225/864, ~245/216
Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}


[[POTE generator]]: ~5/4 = 383.256
Optimal tunings:  
* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}


{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
{{Optimal ET sequence|legend=0| 41, 91 }}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
Badness (Sintel): 0.955


[[Badness]]: 0.0408
== Merman ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''


Scales: [[wizard22]]
Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.


== 11-limit ==
The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 385/384, 4000/3993
[[Comma list]]: 225/224, 2500000/2470629


Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
: mapping generators: ~2, ~10/7


Mapping generators: ~99/70, ~25/22
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}


POTE generator: ~5/4 = 383.232
{{Optimal ET sequence|legend=1| 41, 84, 125 }}


{{Val list|legend=1| 22, 50, 72, 166, 238c, 310c }}
[[Badness]] (Sintel): 1.39


Badness: 0.0185
=== 11-limit ===
Subgroup: 2.3.5.7.11


Scales: [[wizard22]]
Comma list: 225/224, 441/440, 1344/1331


=== Lizard ===
Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}


Subgroup: 2.3.5.7.11.13
Optimal tunings:  
* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}


Comma list: 225/224, 351/350, 364/363, 385/384
{{Optimal ET sequence|legend=0| 41, 84, 125e }}


Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
Badness (Sintel): 1.20


Mapping generators: ~99/70, ~25/22
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~5/4 = 383.389
Comma list: 144/143, 225/224, 364/363, 441/440


{{Val list|legend=1| 22, 50, 72, 122, 194df }}
Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}


Badness: 0.0218
Optimal tunings:  
* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}


==== 17-limit ====
{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}


Subgroup: 2.3.5.7.11.13.17
Badness (Sintel): 1.14


Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
=== Mermaid ===
Subgroup: 2.3.5.7.11


Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
Comma list: 225/224, 385/384, 532400/531441


Mapping generators: ~17/12, ~17/15
Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}


POTE generator: ~5/4 = 383.381
Optimal tunings:  
* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}


{{Val list|legend=1| 22, 50, 72, 122g, 194dfg }}
{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}


Badness: 0.0145
Badness (Sintel): 1.46


</div></div>
==== 13-limit ====
 
Subgroup: 2.3.5.7.11.13
==== 19-limit ====


Subgroup: 2.3.5.7.11.13.17.19
Comma list: 225/224, 325/324, 385/384, 10648/10647


Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}


Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
Optimal tunings:  
* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}


Mapping generators: ~17/12, ~17/15
{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}


POTE generator: ~5/4 = 383.477
Badness (Sintel): 1.47


{{Val list|legend=1| 22h, 50, 72, 122g, 194dfg }}
== Slender ==
Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.


Badness: 0.0157
The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.  


</div></div>
[[Subgroup]]: 2.3.5.7


=== Gizzard ===
[[Comma list]]: 225/224, 589824/588245


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
: mapping generators: ~2, ~49/48


Comma list: 225/224, 325/324, 385/384, 1573/1568
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}


Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}


Mapping generators: ~99/70, ~25/22
[[Badness]] (Sintel): 1.44


POTE generator: ~5/4 = 383.170
=== 11-limit ===
Subgroup: 2.3.5.7.11


{{Val list|legend=1| 72, 166, 238cf }}
Comma list: 225/224, 385/384, 1331/1323


Badness: 0.0203
Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}


</div></div>
Optimal tunings:
* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}


==== 17-limit ====
{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}


Subgroup: 2.3.5.7.11.13.17
Badness (Sintel): 0.838


Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
Comma list: 225/224, 275/273, 385/384, 1331/1323


Mapping generators: ~17/12, ~17/15
Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}


POTE generator: ~5/4 = 383.175
Optimal tunings:  
* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}


{{Val list|legend=1| 72, 166g, 238cfg }}
{{Optimal ET sequence|legend=0| 31, 63, 94 }}


Badness: 0.0136
Badness (Sintel): 1.07


</div></div>
== Triton ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''


==== 19-limit ====
Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.


Subgroup: 2.3.5.7.11.13.17.19
[[Subgroup]]: 2.3.5.7


Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
[[Comma list]]: 225/224, 1029/1000


Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
: mapping generators: ~2, ~10/7


Mapping generators: ~17/12, ~17/15
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}


POTE generator: ~5/4 = 383.138
{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}


{{Val list|legend=1| 72, 94, 166g }}
[[Badness]] (Sintel): 1.50
 
Badness: 0.0148
 
== Mage ==


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 1331/1296
Comma list: 45/44, 56/55, 1029/1000


Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}


Mapping generators: ~77/54, ~55/48
Optimal tunings:
* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}


POTE generator: ~5/4 = 383.124
{{Optimal ET sequence|legend=0| 2, 17d, 19 }}


{{Val list|legend=1| 22, 50e, 72ee, 94ee }}
Badness (Sintel): 1.51


Badness: 0.0578
== Marvolo ==
[[Subgroup]]: 2.3.5.7


= Triton =
[[Comma list]]: 225/224, 156250000/155649627


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
: mapping generators: ~2, ~21/20


Subgroup: 2.3.5.7
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}


[[Comma list]]: 225/224, 1029/1000
{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}


[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
[[Badness]] (Sintel): 2.11


{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


[[POTE generator]]: ~7/5 = 568.865
Comma list: 225/224, 441/440, 4000/3993


{{Val list|legend=1| 19, 78bd, 97bd }}
Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}


[[Badness]]: 0.0592
Optimal tunings:  
* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}


== 11-limit ==
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


Subgroup: 2.3.5.7.11
Badness (Sintel): 0.958


Comma list: 45/44, 56/55, 1029/1000
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
Comma list: 169/168, 225/224, 364/363, 441/440


POTE generator: ~7/5 = 569.144
Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}


{{Val list|legend=1| 19, 59bde, 78bde, 97bde }}
Optimal tunings:
* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}


Badness: 0.0457
{{Optimal ET sequence|legend=0| 29, 43, 72 }}


= Tritonic =
Badness (Sintel): 0.887


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7


[[Comma list]]: 225/224, 50421/50000
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440


[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 0 -5 11 12 }}]
Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}


{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
Optimal tunings:
* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}


[[POTE generator]]: ~7/5 = 580.286
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
Badness (Sintel): 0.760


== 11-limit ==
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Subgroup: 2.3.5.7.11
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440


Comma list: 121/120, 225/224, 441/440
Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}


Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
Optimal tunings:  
* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}


POTE generator: ~7/5 = 580.267
{{Optimal ET sequence|legend=0| 29g, 43, 72 }}


{{Val list|legend=1| 29, 31, 60e }}
Badness (Sintel): 0.895


Badness: 0.0237
== Enneaportent ==
[[Subgroup]]: 2.3.5.7


=== 13-limit ===
[[Comma list]]: 225/224, 40353607/40310784


Subgroup: 2.3.5.7.11.13
{{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
: mapping generators: ~2592/2401, ~12005/6912


Comma list: 105/104, 121/120, 196/195, 275/273
[[Optimal tuning]]s:  
* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}


Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 0 -5 11 12 3 18 }}]
{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}


POTE generator: ~7/5 = 580.108
[[Badness]] (Sintel): 2.37


{{Val list|legend=1| 29, 31, 60e, 151cde }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Badness: 0.0230
Comma list: 225/224, 385/384, 12005/11979


== Tritoni ==
Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})


Comma list: 225/224, 385/384, 27783/27500
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
Badness (Sintel): 1.01


POTE generator: ~7/5 = 580.389
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


{{Val list|legend=1| 31, 91, 122, 153d }}
Comma list: 169/168, 225/224, 364/363, 1716/1715


Badness: 0.0455
Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}


= Merman =
Optimal tunings:
* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}


Subgroup: 2.3.5.7
Badness (Sintel): 0.922


[[Comma list]]: 225/224, 2500000/2470629
== Gracecordial ==
: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''


[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
[[Comma list]]: 225/224, 781250000/771895089


[[POTE generator]]: ~7/5 = 585.585
{{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
: mapping generators: ~2, ~3


{{Val list|legend=1| 41, 84, 125 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}


[[Badness]]: 0.0551
{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}


== 11-limit ==
[[Badness]] (Sintel): 2.44


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 1344/1331
Comma list: 225/224, 385/384, 236328125/234365481


Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}


POTE generator: ~7/5 = 585.606
Optimal tunings:  
* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}


{{Val list|legend=1| 41, 84, 125e }}
{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}


Badness: 0.0364
Badness (Sintel): 2.96


== 13-limit ==
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 385/384, 831875/830466


Comma list: 144/143, 225/224, 364/363, 441/440
Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}


Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
Optimal tunings:  
* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}


POTE generator: ~7/5 = 585.657
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


{{Val list|legend=1| 41, 84, 125e, 209ef, 293ef }}
Badness (Sintel): 2.16


Badness: 0.0275
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17


= Septimin =
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913


: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}


[[Comma list]]: 225/224, 84035/82944
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
Badness (Sintel): 1.96


{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


[[POTE generator]]: ~7/6 = 263.632
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444


{{Val list|legend=1| 41, 91, 132 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}


== 11-limit ==
Optimal tunings:
* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}


Subgroup: 2.3.5.7.11
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}


Comma list: 225/224, 385/384, 2401/2376
Badness (Sintel): 1.71


Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23


POTE generator: ~7/6 = 263.634
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528


{{Val list|legend=1| 41, 91, 223cdef }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}


== 13-limit ==
Optimal tunings:
* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}


Comma list: 105/104, 144/143, 196/195, 245/242
Badness (Sintel): 1.57


Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


POTE generator: ~7/6 = 263.700
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493


{{Val list|legend=1| 41, 91 }}
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}


= Slender =
Optimal tunings:
* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}


Subgroup: 2.3.5.7
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


[[Comma list]]: 225/224, 589824/588245
Badness (Sintel): 1.50


[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
==== 31-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31


{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493


[[POTE generator]]: ~49/48 = 38.413
Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}


{{Val list|legend=1| 31, 94, 125 }}
Optimal tunings:
* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}


[[Badness]]: 0.0569
{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}


== 11-limit ==
Badness (Sintel): 1.53


=== Gracecord ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 1331/1323
Comma list: 225/224, 441/440, 109375/107811


Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}


POTE generator: ~49/48 = 38.387
Optimal tunings:  
* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}


{{Val list|legend=1| 31, 63, 94, 125 }}
{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}


Badness: 0.02534
Badness (Sintel): 2.21
 
== 13-limit ==


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 225/224, 275/273, 385/384, 1331/1323
Comma list: 225/224, 364/363, 441/440, 6125/6084


Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}


POTE generator: ~49/48 = 38.314
Optimal tunings:  
* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}


{{Val list|legend=1| 31, 63, 94 }}
{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}


Badness: 0.02591
Badness (Sintel): 1.83


= Marvo =
==== 17-limit ====
{{see also| Gravity family }}
Subgroup: 2.3.5.7.11.13.17


Subgroup: 2.3.5.7
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989


Comma list: 225/224, 78125000/78121827
Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}


Mapping: [{{val| 1 5 12 29 }}, {{val| 0 -6 -17 -46 }}]
Optimal tunings:  
* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}


{{Multival|legend=1| 6 17 46 13 56 59 }}
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


POTE generator: ~27/20 = 516.694
Badness (Sintel): 1.87


{{Val list|legend=1| 65d, 72, 137, 209, 281, 569bcc }}
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19


Badness: 0.0976
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663


== 11-limit ==
Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}


Subgroup: 2.3.5.7.11
Optimal tunings:  
* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}


Comma list: 225/224, 243/242, 4000/3993
{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}


Mapping: [{{val| 1 5 12 29 12 }}, {{val| 0 -6 -17 -46 -15 }}]
Badness (Sintel): 1.68


POTE generator: ~27/20 = 516.699
== Alphorn ==
[[Subgroup]]: 2.3.5.7


{{Val list|legend=1| 65d, 72, 281, 353c, 425bc, 497bc }}
[[Comma list]]: 225/224, 5764801/5668704


Badness: 0.0317
{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
: mapping generators: ~2, ~35/24


== 13-limit ==
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}


Subgroup: 2.3.5.7.11.13
{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}


Comma list: 225/224, 243/242, 351/350, 1625/1617
[[Badness]] (Sintel): 3.27


Mapping: [{{val| 1 5 12 29 12 39 }}, {{val| 0 -6 -17 -46 -15 -62 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~27/20 = 516.730
Comma list: 225/224, 385/384, 12250/11979


{{Val list|legend=1| 65d, 72, 137, 209, 281f, 490bcf }}
Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}


Badness: 0.0269
Optimal tunings:  
* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}


= Marvolo =
{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}


Subgroup: 2.3.5.7
Badness (Sintel): 2.43


[[Comma list]]: 225/224, 156250000/155649627
== Misneb ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''


[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
[[Comma list]]: 225/224, 4194304/4117715


[[POTE generator]]: ~21/20 = 83.348
{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
: mapping generators: ~2, ~15/8


{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}


[[Badness]]: 0.0833
{{Optimal ET sequence|legend=1| 21, 32, 53 }}


== 11-limit ==
[[Badness]] (Sintel): 3.57


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 441/440, 4000/3993
Comma list: 99/98, 176/175, 1310720/1294139
 
Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]


POTE generator: ~21/20 = 83.340
Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}


{{Val list|legend=1| 29, 43, 72 }}
Optimal tunings:
* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}


Badness: 0.0290
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


</div></div>
Badness (Sintel): 2.82
 
== 13-limit ==


==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 364/363, 441/440
Comma list: 99/98, 176/175, 640/637, 847/845
 
Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]


POTE generator: ~21/20 = 83.330
Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}


{{Val list|legend=1| 29, 43, 72, 115f }}
Optimal tunings:
* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}


Badness: 0.0215
{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}


</div></div>
Badness (Sintel): 1.88


= Amavil =
=== Musneb ===
== 5-limit (mabila) ==
Subgroup: 2.3.5.7.11


Subgroup: 2.3.5
Comma list: 225/224, 385/384, 66550/64827


[[Comma list]]: 268435456/263671875
Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}


[[Mapping]]: [{{val| 1 6 1 }}, {{val| 0 -10 3 }}]
Optimal tunings:  
* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}


[[POTE generator]]: ~512/375 = 529.6849
{{Optimal ET sequence|legend=0| 21e, 32, 53 }}


{{Val list|legend=1| 9, 25, 34, 77, 111, 145, 256c }}
Badness (Sintel): 2.89


[[Badness]]: 0.2325
== Untriton ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''


== 7-limit ==
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 17496/16807
[[Comma list]]: 225/224, 125000000/121060821


[[Mapping]]: [{{val| 1 6 1 9 }}, {{val| 0 -10 3 -14 }}]
{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
: mapping generators: ~2, ~10/7


{{Multival|legend=1| 10 -3 14 -28 -6 41 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}


[[POTE generator]]: ~48/35 = 529.979
{{Optimal ET sequence|legend=1| 51, 53 }}


{{Val list|legend=1| 9, 25d, 34d, 43, 77d }}
[[Badness]] (Sintel): 3.64
 
[[Badness]]: 0.1096
 
== 11-limit ==


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 99/98, 176/175, 864/847
Comma list: 121/120, 225/224, 22000/21609


Mapping: [{{val| 1 6 1 9 7 }}, {{val| 0 -10 3 -14 -8 }}]
Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}


POTE generator: ~15/11 = 529.974
Optimal tunings:  
* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}


{{Val list|legend=1| 9, 34d, 43, 77de }}
{{Optimal ET sequence|legend=0| 51, 53 }}


Badness: 0.0426
Badness (Sintel): 2.46
 
== 13-limit ==


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 78/77, 99/98, 144/143, 176/175
Comma list: 121/120, 225/224, 275/273, 1040/1029
 
Mapping: [{{val| 1 6 1 9 7 9 }}, {{val| 0 -10 3 -14 -8 -12 }}]
 
POTE generator: ~15/11 = 529.951


{{Val list|legend=1| 9, 34d, 43, 77de }}
Mapping: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}


Badness: 0.0258
Optimal tunings:  
* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}


= Enneaportent =
{{Optimal ET sequence|legend=0| 51f, 53 }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.96


[[Comma list]]: 225/224, 40353607/40310784
== Naiadical ==
Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.


[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
[[Comma list]]: 225/224, 823543/800000


[[POTE generator]]: ~5/4 = 383.165
{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
: mapping generators: ~2, ~32/21


{{Val list|legend=1| 63, 72, 495bcd }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}


[[Badness]]: 0.0937
{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}


== 11-limit ==
[[Badness]] (Sintel): 3.67


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 12005/11979
Comma list: 225/224, 245/242, 1617/1600


Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}


POTE generator: ~5/4 = 383.146
Optimal tunings:  
* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}


{{Val list|legend=1| 63, 72, 423cd, 495bcd }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0304
Badness (Sintel): 2.00
 
</div></div>
 
== 13-limit ==


=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 364/363, 1716/1715
Comma list: 105/104, 196/195, 245/242, 1001/1000


Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}


POTE generator: ~5/4 = 383.047
Optimal tunings:  
* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}


{{Val list|legend=1| 63, 72, 279cf }}
{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}


Badness: 0.0223
Badness (Sintel): 1.43


</div></div>
=== 17-limit ===
 
Subgroup: 2.3.5.7.11.13.17
= Submajor =
== 5-limit ==
 
Subgroup: 2.3.5
 
[[Comma list]]: 69198046875/68719476736
 
[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]


[[POTE generator]]: ~10125/8192 = 362.321
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242


{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc}}
Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}


[[Badness]]: 0.1302
Optimal tunings:  
* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}


== 7-limit ==
{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.26


[[Comma list]]: 225/224, 51200/50421
== Quintannic ==
Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.


[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
[[Comma list]]: 225/224, 9805926501/9765625000


[[POTE generator]]: ~49/40 = 362.255
{{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
: mapping generators: ~2, ~10000/9261


{{Val list|legend=1| 10, 33, 43, 53 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}


[[Badness]]: 0.0605
{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}


== 11-limit ==
[[Badness]] (Sintel): 3.81


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 6655/6561
Comma list: 225/224, 441/440, 43923/43750


Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}


POTE generator: ~27/22 = 362.101
Optimal tunings:  
* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}


{{Val list|legend=1| 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}


Badness: 0.0506
Badness (Sintel): 1.74


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 225/224, 275/273, 385/384
Comma list: 225/224, 441/440, 1001/1000, 1188/1183


Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}


POTE generator: ~16/13 = 362.105
Optimal tunings:  
* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}


{{Val list|legend=1| 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Badness: 0.0277
Badness (Sintel): 1.35


</div></div>
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


== Interpental ==
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}


Comma list: 99/98, 176/175, 51200/50421
Optimal tunings:  
* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}


POTE generator: ~49/40 = 362.418
{{Optimal ET sequence|legend=0| 43, 60e, 103 }}


Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
Badness (Sintel): 1.17


POTE generator: ~49/40 = 362.418
== Gwazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''


{{Val list|legend=1| 43, 53, 96, 149d }}
Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.


Badness: 0.0518
[[Subgroup]]: 2.3.5.7


</div></div>
[[Comma list]]: 225/224, 5971968/5764801


=== 13-limit ===
{{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
: mapping generators: ~2401/1728, ~35/32


Subgroup: 2.3.5.7.11.13
[[Optimal tuning]]s:  
* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}


Comma list: 99/98, 169/168, 176/175, 640/637
{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}


POTE generator: ~16/13 = 362.402
[[Badness]] (Sintel): 4.53


Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


POTE generator: ~16/13 = 362.402
Comma list: 99/98, 176/175, 65536/65219


{{Val list|legend=1| 43, 53, 96, 149d }}
Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}


Badness: 0.0297
Optimal tunings:  
* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}


</div></div>
{{Optimal ET sequence|legend=0| 22, 74, 96 }}


= Alphorn =
Badness (Sintel): 2.26


Subgroup: 2.3.5.7
== Tertiosec ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''


[[Comma list]]: 225/224, 5764801/5668704
Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.


[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 0 -16 5 -22 }}]
[[Subgroup]]: 2.3.5.7


{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
[[Comma list]]: 225/224, 14495514624/13841287201


[[POTE generator]]: ~48/35 = 556.221
{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
: mapping generators: ~3072/2401, ~2048/1715


{{Val list|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
[[Optimal tuning]]s:
* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}


[[Badness]]: 0.1293
{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}


== 11-limit ==
[[Badness]] (Sintel): 10.9


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 225/224, 385/384, 12250/11979
Comma list: 225/224, 3840/3773, 12005/11979
 
Mapping: {{mapping| 3 -1 12 7 14 | 0 8 -7 2 -5 }}


Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
Optimal tunings:  
* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}


POTE generator: ~11/8 = 556.144
{{Optimal ET sequence|legend=0| 21, 54, 75e }}


{{Val list|legend=1| 28d, 41, 315cde }}
Badness (Sintel): 5.74


Badness: 0.0735
== References ==


[[Category:Regular temperament theory]]
[[Category:Temperament collections]]
[[Category:Temperament collection]]
[[Category:Marvel temperaments| ]] <!-- main article -->
[[Category:Marvel]]
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Marvel_temperaments"